Hodge theory and deformations of maps

C ^OMPOSITIO M ^ATHEMATICA

Z ^IV R ^AN

Hodge theory and deformations of maps

Compositio Mathematica, tome 97, n

^o

3 (1995), p. 309-328

<http://www.numdam.org/item?id=CM_1995__97_3_309_0>

© Foundation Compositio Mathematica, 1995, tous droits réservés.

L’accès aux archives de la revue « Compositio Mathematica » (http:

//http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

Hodge theory ^and deformations of _maps

ZIV RAN*

Department of Mathematics, University of California, Riverside, CA 92521

Received 23 January 1992; accepted in revised form 30 June 1993; final revision June 1995.

0 1995 Kluwer Academic Publishers. Printed

The purpose of this paper is ^to establish and

apply

^a

general principle (Theorem ^1.1,

^Section

1)

^{which serves to relate}

Hodge theory

^{and defor-}

mation

theory.

^This

principle,

^{which we state in a} somewhat technical abstract-nonsense

framework,

^{takes on two} dual forms which _say

roughly, respectively,

^that:

(0.1) for a functorial homomorphism

from

^a

Hodge-type

group ^{to an}

infinitesimal deformation

group,

im( 1")

^consists

of

unobstructed

deformations;

(0.2) for a functorial homomorphism

from

^an obstruction _group to a

Hodge-type

group, "actual"

obstructions lie in

ker( n).

A familiar

example of (0.1)

^{is the case} of deformations of manifolds with trivial canonical

bundle,

^{treated from a similar}

viewpoint

ⁱⁿ

[Rl] ;

^a

familiar

example

^of

(0.2)

^{is Bloch’s}

semi-regularity

map

[B],

treated from

a similar

viewpoint

ⁱⁿ

[R3].

^Our

principle

may be viewed as an

amalga-

mation and abstraction of parts ^of

[R1]

^and

[R3].

We will

apply

^{the above}

principle

^to deformations of _maps of

complex

manifolds. After

recalling

^and

developing

^{in Section 2 some}

general

formalism

concerning

^such

deformations,

^we

develop,

in Sections 3 and

4,

two instances of a map 03C0 ^as in

(0.2)

^{in the}

respective

^{cases of}

generic

immersions and fibre spaces. It may be noted that even in the case of an

*Supported ⁱⁿ part by ^{NSF. DMS 9202050.}

embedding

^our map 03C0 does not reduce to Bloch’s

semi-regularity

map but rather has the latter as a component

(cf.

^Remark

3.2.1);

ⁱⁿ

particular

^our

map has smaller kernel than

Bloch’s, leading

^{to a better} consequence of

(0.2) (cf.

^Remark

3.4.1).

Some

applications given

in Sections 3 and 4 are to the

question

^of

stability,

^{i.e. of}

identifying, given

^a

generic

^immersion

f :

^{Y - X or a fibre}

space

f: X - x

^the locus of deformations of X to which

f

^{extends. We will}

show,

ⁱⁿ

particular,

that under favorable conditions the latter locus _{may be} identified

Hodge-theoretically (Corollaries ^3.2, 4.2);

^{in the case of an}

embedding f

^this may be considered as a

special

^{case of an} infinitesimal form of the Generalized

Hodge Conjecture.

^A

particularly

^{nice case is that}

of

q-Lagrangian

submanifolds of a

q-symplectic manifold,

^{in which we}

give

a result

(Corollary 3.4) generalizing

^{work of Voisin}

[V].

In the final Section 5 we

systematically apply

^the

foregoing

^{to the}

problem

^of

moving

curves on a

manifold,

^a

problem

^made

popular

ⁱⁿ

recent years

by

^Mori

theory.

^{We will}

give

^a

variety

of conditions under which a curve is forced to move. As a final

application,

^{we describe the}

structure of manifolds X such that

Qi

^is

spanned

^{off a finite set:}

namely

^we

prove

(Theorem 5.3)

^{that either X is fibred}

by

^{tori or the} canonical bundle

Kx

^is

ample.

^{This is in}

complete analogy

^{with the known case of 1-forms}

(Ueno’s conjecture,

^cf.

[R4]).

^{It is}

interesting

^{to note that the}

proof

^involves

applications

^{of both}

(0.1)

^and

(0.2) (at

^{two different}

points).

We

begin

in Section 1

by setting

up ^an abstract

categorical

^framework

for deformation

theory,

^{one which seems} best suited for

stating (let

^alone

proving)

^our

general principle;

it is shown in Section 2 that deformations of _maps of manifolds fit into this

framework,

^{which is a variant} of the usual

one

(cf. [S]); perhaps

^{the main}

novelty

^{here over}

[S]

^{is the}

systematic

^use

of "canonical elements" to describe infinitesimal

deformations,

ⁱⁿ

general-

ization of the method of

[Rl]

^and

[R2].

^{For a different}

application

^{of the}

results of Section

1,

^see

[R6].

Convention. In this _paper ’Kahler manifold’ should be understood in the

cohomological

sense, i.e. a manifold whose

Hodge-de

^Rham

spectral

sequence

degenerates

^at

E 1.

1. Abstract nonsense

Let A be the category ^{of Artin local}

k-algebras

^with residue field k.

In order to use

geometric language,

^we may,

by

^the

assignment

R - S ⁼

Spec(R), identify

^the

opposite

category ^{dOP with a} category ^of

(1-point)

^{schemes over} k. We may

also,

^{from time to}

time,

^{refer to an}

element R ~ A as

(R, M)

^or

(R, M, I)

^{where M} is the maximal ideal and

I ⁼

Ann(M)

^{ci R. To} category ^{such as A we} may associate 2 other

categories:

W-.,Yom whose

objects

^are

morphisms

^{of A and whose}

morphism

^are commutative squares in

A;

^and

A-Mod,

^whose

objects

^are

pairs (R, A)

^where

R ~ A,

^{and A is an}

R-module;

^{and whose}

morphisms

^are

defined

by

Now consider a category

AIW,

^{i.e. a small} category

together

^{with a}

contravariant functor 03A6:B ^-

.91;

^we will sometimes write an

object

^{X of 11}

as

X/R

^or

X/S indicating

^{that R =}

O(X)

^{or S =}

Spec(03A6(X)).

^{We will}

assume that 11 has an

unique object X °/k.

Define another

category éé

^as

the fibre

product

We will say that the category

14/.91

^is

base-changing

^{if it comes}

equipped

with a

functor À -

e-Yeom such that the

following diagram

^commutes

where

q’(X2 ~ X 1)

⁼

X1

and 03A6’ is the evident functor induced

by 03A6;

^more

concretely,

^{what this means is that any}

X1/S1

^and

S2 ~ SI

may be

functorially completed

^{to a "base}

change" diagram

In this case we will use the notation

X 2/S2

⁼

X

1

S1 S2, X21R,

⁼

X

1 XR1

R2,

Si

⁼

Spec(Ri).

Note that to a

base-changing

category

B/A

^{we may associate a}

(covariant)

functor

and we will _say that A is a realization of F. In what follows we will assume

that F admits a hull in the sense of

Schlessinger [S],

^{in which case we will} say that A is hullable.

Let à

⁼

{(X/R, R’):X/R~B, R

⁼

R’/FR’ = R’/F’}.

Now

given B/A

^as

above, by

^an admissible

B-module,

^{we mean a functor}

L:

à -+

^A-Mod

compatible

^with

03A6, together

^{with the}

data,

^{for all}

(R, M, I)

^{E d}

and

(X/R, R’) ~ ,

^{of a} functorial exact

diagram

where the top ^{row is the obvious}

thing and g

is the map induced

by base-change, plus

^a group

L(X’)

with functorial

isomorphisms,

^{for all}

(R, M, I

⁼

M)

~ A,

L(X0/k, R)

⁼

L(X0)

^{(D M. L is said to be}

right (resp. left)-

exact

if g

^is

always surjective (resp. f

^is

always injective). By

^a

ô-pair of

A-modules we mean a

pair (LI, L2)

of admissible A-modules

together

^{with the}

data,

^{for all}

(R, M, I)~A

^and

X/R~B,

^{of a} functorial exact sequence

L1(XO)

^{(D 1}

~ L1(X, R’)

^~

L1(X x (R/1), R) ~ L2(X0)

^{O 1 ~}

L2(X, R’)

~ L2(X x ^{(R/1), R).}

By

^a linearization of

PÃ / .91

^{we mean a}

ô-pair (L’, L2)

of admissible A-modules

together

^{with the}

data,

^{for all}

(R, M, I) ~ A, J

⁼

Ann(M/I)

^{c R and}

X/(R/1) ^{~ B,}

^{of a} canonical element

with the property

that X

comes via

base-change

^{from some}

X/R

^{iff a lies in}

the

image

^{of the canonical} map

’Vote that

given

^a linearized

base-changing

category

B/A,

^{an admissible}

module L’ ^c LI

gives

^{rise to a}

subcategory

^{é8’ c: A defined}

"inductively" by

the conditions that

X0/K~B’

^{and that}

given X/R ~ B

^{such that X}

x R (R/1) ^{G é3’,} X/R ~ B’

iff the canonical element

Next, by

^{an exact}

triple

of the linearized

base-changing categories

^{over si}

we mean a

triple (Bi, Li , L2i)

^of

such, together

with functors

an extension

of (L11, L12)

^as

ô-pair

^to

PÃ2,

still denoted

(L11, L21),

^{and an exact}

sequence of

PÃ2-modules

where

(L13, L23)

^{are viewed as}

e2-modules

^{in the obvious} way, such that the

respective

canonical a-elements are

compatible.

Now ^{we can}

finally

^{state our result:}

THEOREM 1.1. Let

(Bi, Ll, L?),

^{i =}

1, 2,

^{3 be an exact}

triple of

^linearized

base-changing .91 -categories, realizing respectively

^the

functors Fi:

^{A ~ Yets.}

Let

Z,,

^{i =}

^{1, 2, 3,}

^{be a}

formal

scheme which is a

hull for Fi

^and

Z1 ~ Z2 ~ Z3

the induced _maps. Also put

lji

⁼

dimk(Lji(X0i))

^where

X0i/k~Bi

^{is the}

unique object.

(i) Suppose

^{K is a}

right-exact B3-module

^{with a natural}

transformation

and put ^{Q =}

rkk(03C4: K(X03) ~ L13(X03)).

^{Then there is a} smooth a-dimensional

formal

^subscheme

Zo

^c

Z3 corresponding

^to

im(r),

^{such that}

Zo:= 03B2-12(Z03)

^has

embedding

dimension ^{6 +}

l11 and

^is

defined by

^{at most}

11 equations.

(ii) Suppose

^{K is a}

left-exact A2-module

^{with a natural}

transformation

and put p ⁼

rkk(03C0: L21(X01) ~ K(X01)). Suppose Z03

^c

Z3 is a formal

^subscheme

corresponding to a subfunctor of

Then the natural _map

Z’ = 03B2-12(Z03) ~ Z’factors

where B is smooth

of dimension if

^{and i is an}

embedding

^onto

a formal

^subscheme

defined by

^{at most}

11

^- p

equations.

Proof.

^Cases

(i)

^and

(ii)

^are

essentially

^{dual to each}

other;

^{moreover the}

proof

^{of case}

(ii)

is almost word-for-word the same as that of Theorem 1 in

[R3],

^{which is its}

special

^case

concerning

deformations of

embeddings

^and

Bloch’s

semi-regularity

map

(cf. §2, 3).

^For

completeness’ sake,

^we will sketch the

proof

^{of case}

(i),

^{which in turn is similar to that} of Theorem 1 in

[R1].

Take

(R, M, I) ~ A,

^{J =}

Ann(M/1),

^and suppose ^we have

such that

X3 ^belongs

^{to the}

subcategory Ao 3

^c

B3 corresponding

^to

Z’, 3

^i.e.

such that the canonical element

lies in the

image

^of

Then the

right-exactness

^{of K}

yields immediately

^that

a3

^{lifts to an element}

03B13 ~ im(K(X3, R) ~ L13(X3, R)),

^hence

X3

^{lifts to some}

X3/R ~ B03.

^{This shows}

that

Z03

^is

smooth,

^{and to}

complete

^the

proof

it remains to

investigate

^the

obstruction to

lifting X2

^{to some}

X2/R ~ B2

which maps ^to

X3,

^or what is the same, the obstruction to

lifting

the canonical element

to an element

(1.2 E Li(X 2’ R)

which maps ^to a3. Now from the commutative

diagram

it is clear that the latter obstruction lies in I Q

L21(X02).

^D

2. Generalities ^on déformations of _maps

In

[R5],

^{we described a}

general

formalism for

studying

deformations of _maps.

Our _purpose here is to revisit and elaborate on this in the

relatively benign

case of _maps of

manifolds, fitting it,

ⁱⁿ

particular,

into the abstract framework of Section 1. We

begin

^{with some}

topology.

^Let

be ^a continuous _map of

topological

spaces. As in

[R5],

^{we associate to}

f

^a

Grothendieck

topology 03C4(f)

whose open ^{sets are} the

pairs (U, V)

^{such that}

U c X and V ^c Y are open and

f(U)

c V, and whose notion of

covering

^is

the obvious one. Note that we have a commutative

diagram

^{of continuous}

maps of Grothendieck

topologies

where i,

j, n

^are

defined, respectively, by

For sheaves E on X and F

on Y,

^{we define sheaves}

F#

⁼ sheafification of 1

Note that we have

and that there is a canonical

diagram

of sheaves on

i(f):

where f#

^is

injective provided f

^{is open. Put}

A small remark which will be useful later is

LEMMA 2.1.

Suppose f is

open and

f* f-1F

^{F. Then we have}

Proof.

^{Consider the exact sheaf sequence on}

03C4(f)

Applying

03C0* ^to

(2.2), using

^{the basic}

diagram (2.1)

^{and our}

hypothesis

^on

F, ^{we conclude}

easily

^that

Hence the

Leray spectral

sequence for

QF

^{and 7c}

yield

^{the Lemma.}

Suppose

^{now that our}

f :

^{X ~ Y is a map of}

complex

manifolds. We may then define on

T(f )

^{a sheaf}

Tf

^of

’f-related pairs

^of germs of vector fields’

by

Note the exact sequence

and its

cohomology

sequence

where

Now denote

by Af (resp. é0x, é0y)

^the category of deformations of

f (resp. X, Y).

^{Based on} the results of

[R7],

^{we extend}

Tl , T?

^{to functors on}

Ax making

up ^a

a-pair,

^{in the}

following

way,

assuming H0(0398X)

^{= 0. Let}

J · (TX)

⁼

lim Jm(TX)

be the Jacobi

complex

associated to the Lie

algebra TX

^and

^{R univ =} lim Runivm

^the

ring

of the universal formal deformation. Given

(X1/R, R’) ~ X,

^{there is a}

Kodaira-Spencer homomorphism

^{R univ - R}

which endows

MR,

^{and hence}

T(R’ )

⁼

Mi,

^{with an}

^{Runiv -}

^{module struc-}

ture, which

obviously

^factors

through Runivm, m

⁼

exponent(R).

^Define

The canonical element is

just

^the

Kodaira-Spencer

map TR -

HO(J . (TX)).

Similarly

^for

TiY

^and

Tif.

PROPOSITION 2.2. For _any compact

complex manifold

^X

(resp.

map

f:

^{X - Y}

of such),

the category

BX/A (resp. Bf/A) ^forms

^a linearized

base-changing

category, with associated

ô-pair (T1X, T2) (resp. (T1f, T2f».

Proof.

^{This is a classical save}

possibly

^{for the} part

involving

^{the canonical}

element a. As for the

latter,

^{the case} of manifolds X is discussed in detail in

[R7],

while that of _map

f

^{is similar}

(and

^{is mentioned in}

[R2]).

EXAMPLE 2.2. A

special

^case of Theorem 1.1 is when

A2 = e3

⁼

Bf, 311

^{= .91}

(the

"trivial"

.91 -category).

^{In case}

(i), assuming

^{further that r is an}

isomorphism,

^the conclusion is that

Def( f )

^is

smooth;

^{in other}

words,

^when-

ever

T1f

^is

right-exact (i.e.,

^{in the}

terminology

^of

[Rl], f

^{satisfies the}

T’-Iifting property)

^then

f

^is unobstructed. This is

precisely

^{Theorem 1 of}

[Rl].

3. Generic immersions

We now consider a

generic

^immersion

(i.e. generically

unramified

map) f :

Ym ~ X". Let N denote the normal sheaf

of f,

^defined

by

^{the exact}

sequence ^on Y

Combining (3.1)

^with

(2.3),

^{we obtain the}

following

^exact sequence ^on

03C4(f):

and its

cohomology

sequence

Denote

by efix

^{the category of} deformations of

f

with fixed target

X,

^and

note as before that

H0(N), H’(N)

^{extend to a}

^ô-pair on Bf/X ^{and ex,}

^and

that the

following

^{is clear.}

PROPOSITION 3.1

(i) eflx ^forms

^a linearized

base-changing

category with associated

a-pair

(HO(N), H1(N)).

(ii) Aflx

^-

Bf ~ ^{BX forms}

^{an exact}

^triple.

Our purpose ^now is to define a

Bf-module

^K

^together

^{with a natural}

transformation

so that Theorem

1.1(ii),

^becomes

applicable.

We work in the derived

categories

^{of the}

topologies

ⁱⁿ

question, identifying

^{N via}

(3.1)

^{with the}

(quasi-isomorphism

^class

of)

^the

complex having Ty

ⁱⁿ

degree -1, f -’Tx

in

degree 0,

^{and otherwise zeros. Let}

Ci

^{be the}

^complex

^on

^{03C4(f) given by}

Consider the

tensor-product complex

^{N+ ~}

Cj, ^{given by}

and note the natural "contraction" _map

whence a map ^on

cohomology

Transposing,

^we obtain maps

which we may assemble

together

^as

Note that the

spectral

sequence of

hypercohomology

^of

Cj ^yields

^exact

sequences

where

Hi,i(f ): Hj.i(X)

^-

Hj,i(Y)

^{is the}

pullback

map. In

particular,

^standard

Hodge theory (cf. [D]) implies

^{for X Kâhler that}

extends to a left

(as

^{well as}

right)-exact Bf-module. ^Moreover,

^{note the}

commutative

diagram

This

implies

^the

following.

^Let

be the natural _map induced

by cup-product,

^and

Tl’o

^c

Tl

^the

subgroup

^of

elements

leaving

E9

ker(Hj,i(f))

^invariant

(via ~).

^Then

T1,0X

^{extends to a}

functor on

BX

^which

gives

^{rise to a}

subcategory B0X

^c

BX

which consists

precisely

of the deformations

preserving ker(H ·,· (f))

^as

sub-Hodge

^structure,

and

similarly

^{a formal subscheme}

which may be

thought

^{of as a} type of "infinitesimal Noether-Lefschetz locus".

The

following, then,

^{is clear} from the above discussion:

COROLLARY 3.2. With the above notations, ^Theorem

1.1, (ii)

^is

applicable

^to

the exact

triple Bf/X ~ Af

^-

^Ax,

^the

subcategory B0X

^c

BX

^{and the map}

1-1:

H1(N) ~ K, provided

^{X is} Kâhler. In

particular,

^{the hull}

Def(f/X) of f

^has

embedding

^dimension

hO(N)

^{and is}

defined by

^{at most}

hl(N)-p equations,

p :=

rkc(n).

^{The latter} conclusion holds even

if

^{X is not Kâhler.}

Proof.

^{It remains to}

justify

^{the last sentence. The}

point

^here is that the Kahler

hypothesis

^{on X is used}

only

ⁱⁿ

controlling

^the

Hodge theory

of deformations of

X,

and these do not come into

play

ⁱⁿ

considering Def(f/X).

REMARK 3.2.1. Note that

Cm+

^{1 =}

QX+1. Identifying Hm-1(03A9m+1X)* = Hp+1(03A9p-1X), p

^{= n - m, it} is easy ^{to see} that for

f

^an

embedding,

03C0m-1,m+1 coincides with the

Kodaira-Spencer-Bloch semi-regularity

map. Thus Corol-

lary

3.2 constitutes both a refinement and a

generalization

of Theorem 1 of

[R3].

In section

5,

^{we will}

apply Corollary

^3.2

systematically

^{to curves. For now}

we

give

^{2 other}

applications.

^{First to an} infinitesimal form of the

generalized Hodge conjecture:

this would _say that in the above situation we should have

In this

regard, Corollary

^3.2

yields

^the

following.

COROLLARY 3.3. In the situation

of Corollary 3.2,

suppose that

and X is Kâhler. Then the

infinitesimal generalized Hodge conjecture (3.3)

^holds.

Next,

^{let us} say that ^a manifold X" is

q-symplectic

^{if it admits a}

holomorphic q-form

^{w such} that the induced contraction _map 1 eu:

TX ~ 03A9q-

^{1X is a bundle}

injection;

^{in this case an immersion}

f:

^{Ym ~ X is said to be}

q-lagrangian

^{if the}

pullback

^{of eu on Y} vanishes and m

+ m - n _{(if q}

^{= 2} these become the usual notions of

complex symplectic

manifold and

lagrangian submanifold).

Note that for

such Y,

^{eu induces an}

isomorphism

^{N -:+.}

03A9q-1Y,

^{and in}

particular

the _map no, ^is

injective,

^{hence so} is n. Hence

Corollary

^3.2

yields

COROLLARY 3.4. Let X be a

q-symplectic complex manifold

^and

f:

^{Y ~ X a}

q-lagrangian

immersion. Then

Def(f/X)

^{is smooth.}

If moreover

^{X is Kâhler then}

the

infinitesimal generalized Hodge conjecture (3.3)

^holds.

REMARK 3.4.1.

For q

^{= 2 this was} proven

by

^Voisin

[V],

^{who also}

points

out that Bloch’s

semi-regularity

map, i.e. 03C0m-1,m+1, need ^{not be}

injective

^{in this}

case, whence the

advantage

^of

using

^n.

4. Fibre _spaces

We now consider the case where our map

f:

^{Xn ~ Ym of} compact

complex

manifolds is a

fibre

space, i.e.

f

is proper, flat and

f* (9x

⁼

(9y. Combining

the exact sequence

(2.3)

^with

(2.2)

^{for F =}

Ty,

^{we obtain the exact sheaf}

sequence ^on

03C4(f)

whose

cohomology

^sequence

^reads,

^{in view of Lemma}

^2.1,

Denote

by BXBf

^{the category} of deformations of

f

^{with fixed source X. Note}

that in this case

BXBf ~ A,

i.e. there are no nontrivial deformations of

f fixing

X, ^{so that the natural}

map Bf ~ BX

^{is in fact an}

embedding,

^hence

Def(f) Def(X).

^As

before,

^the

pair

^of groups

(0, H0(R1 f*OX

^Q

TY))

^extends

to a

ô-pair

^on

BXBf and Bf,

^{and we have the}

^following

PROPOSITION 4.1.

BXBf~ Bf ~ BX ^forms

^{an exact}

^triple.

We

proceed again

^{to define a} map

Define a sheaf

Ci

^on

03C4(f) by

^{the exact sequence}

Note the commutative

diagram

where the left vertical arrow is contraction and the

right

^{vertical arrow is}

deduced in an obvious _way from the

pairing (03A9Y)+

^Q

(f*TY)+ ~ (OX)+

^{and the}

map (03A9j-1Y)+ ~ (03A9j-1X)+.

The

right

^{vertical arrow of}

(4.1)

^then

yields

^a

pairing

^on

cohomology

whence a map

and we may assemble these into

As

before,

^{we have an exact} sequence

where

H·,·(f)

^{is the}

pullback

map,

showing

ⁱⁿ

particular

^{that K}

yields

^a

left-exact module on

Bf.

^{And as before}

^(3.3),

03BCij ^is

compatible

^{with the natural}

composite

map

and we may consider the

subgroup T1,0X

^c

T’ leaving

^{invariant Et)}

im(Hj,i(f)),

and the associated

subcategory B0X

^ce

BX

and subscheme

Def(X)O

^c

Def(X),

which

evidently

^contains

Def(f).

^In

analogy

^with

Corollary 3.2,

^{we have} COROLLARY 4.2. With the above notation, ^Theorem

1.1(ii),

^is

applicable

^to

the exact

triple BXBf ~ Bf ~ ^BX,

^the

subcategory B0X

^c

31 x

^{and the map}

03A3:

H0(R1f*OX

^Q

^Ty)

^{- K. In}

particular,

^{as subscheme}

of Def(X)0, Def(f)

^is

defined by

^{at most}

hO(R 1f*(!J x

^Q

^{Ty) -}

^rk03A3

^equations

^and

^{if Y-}

^is

^injective

^then

Def(f)

⁼

Def(X)0.

REMARK 4.3. It is not hard to check that the target _{of 6m,m} ^is

just

Hm-1,m+

1(X)

^{and that the}

composite

map

represents the obstruction to the

cohomology

^class

[F]

^{of a fibre of}

f remaining

^of type

(m, m)

^under deformation of X; ⁱⁿ

particular,

^whenever _6m,m

is itself

injective,

^the "Noether-Lefschetz" type ^sublocus

Def(X)00

^c

Def(X)

where

[F]

^has type

(m, m) already

^{coincides with}

Def(f).

5.

Moving

^curves

We now consider

applications

^{of the}

preceding results,

ⁱⁿ

particular Corollary

3.2,

^{to the}

problem

^of

moving

^{curves on}

complex

manifolds. Let

f :

^{Y ~ X be}

a nonconstant map from a connected

projective nonsingular

^{curve of} genus g to a

complex

n-manifold

(we

^{call this a curve of} genus g ^on

X)

^{and let H be}

any

component

^of

Def(f/X) through {f}.

^Then

Corollary

^3.2

coupled

^with

Riemann-Roch

yields

the basic estimate

where

Y.KX:= deg(f *Kx)

^and Pi ⁼

rk(03C00i).

^{Note that in the present circum-}

stances

H’(N)

^{is dual to}

H°(N* 0 Qy)

^{and the maps} n01’ 03C002 ^are

respectively

dual to the natural _maps

the last _map

coming

^{from the exact} sequence

We seek some conditions which

imply

that 03C11 or P2 is

positive.

^{To this}

^end,

note the

following elementary

observations:

(i)

^if ri

~ker(H1,0(f))

^{is nonzero at some}

point

^of

f(Y),

^and

0 ~03B6 ~ H0(03A9Y),

then

t03C001(~

⁽⁸⁾

() * 0;

(ii)

^if

coc-H l(Ç12 x)

^is

nondegenerate

^{at some}

point

^of

f(Y),

^then

t03C002(03C9) ~ ^0,

hence 03C12 &#x3E;

0;

(iii)

^if

03A92X

^is

spanned

^{at some}

point

^of

f(Y),

^then

H0(03A92X) generically

spans N* (D

Qy,

^hence

03C12 n - 1;

(iv)

ⁱⁿ

general, 03C12 rk(f*) - h°(A2N*);

^{moreover if n = 3 then A2N* is a}

line bundle on Y of

degree

^{at most v = Y.}

KX - (2g - 2),

^{so its}

^h°

may be estimated.

For

example, combining (ii)

^with

(5.1) yields

^the

following result,

^which

was

already

^{noted in}

[R3]

^{in the case of smooth curves.}

COROLLARY 5.1. Let X be a

symplectic n-fold.

^Then any rational

(resp.

elliptic)

^{curve in X moves in}

a family of dimension

^{at least n - 2}

(resp. 1).

EXAMPLE 5.2. A rational curve on a

symplectic

^{4-fold X must fill} up either a divisor or a rational surface. Consideration of the case

X ⁼

Jeilb2(S),

^{S a K3}

surface,

^{shows that both}

possibilities

can occur:

indeed take Y of the form s + Z where Z ^c S is a rational curve and s e

S;

if s ~ Z

^{then Y fills}

up S

^{+ Z,} while if SE Z then Y fills _up

H ilb2(Z) ~

^P2.

As another

application,

^we consider the structure of manifolds

carrying

many 2-forms. To put ^{matters in}

perspective,

^we recall first the well-known

case of 1-forms: if X is a Kähler manifold such that

n’ x

^is

spanned,

^then

the Albanese

mapping

is an

immersion,

and it is well-known

([U],

^Thm.

^10.9,

^p.

120)

that either X is fibred

by ^tori,

^or else it is of

general

type, ^{in which case} it is known that the canonical bundle

Kx

^is

ample [R4].

THEOREM 5.3. Let X be a Kähler

manifold of

dimension n 3 such that

Qi is spanned

^{outside a}

finite

^set. Then either X is

fibred by

^{tori or its}

canonical bundle K is

ample.

Proof

Note first that

(n - 1)K

⁼

det(03A92X)

^has finite base locus and it follows

by

^some classical work of Zariski

[Z] (kindly pointed

^{out to me}

by

M.

Reid)

^{that mK is free for}

large

^{m, so that we have a stable}

pluricanonical morphism

We

begin

^{with the case n = 3.}

Suppose

first that X has Kodaira dimension x ⁼ 0. As 2K is

effective,

^{2K must be}

trivial,

^{so that we can write}

where L is a

fixed-point-free

involution and K ⁼

O.

^As

T

⁼

03A92

^~

Ki

^{1 =}

03A92

^is

spanned

^{outside a} finite set, it follows from Claim 5.3.0 below

that X

must be a torus; moreover as

H0(03A92X)

⁼

H0(03A92)03C4

^is

^(at

least) 3-dimensional,

^{L must be a} pure

translation,

^{so that X itself is a}

complex

^torus.

CLAIM 5.3.0. Let X be a non-uniruled compact Kahler

manifold

^with

generically spanned

tangent bundle. Then X is a

complex

^torus.

Proof.

^Let

Aut°(X)

dénote the

identity

component ^{of the}

holomorphic automorphism

group of X and

Alb(X)

its Albanese torus. There is a

natural _map

By

results of

Fukiji [F]

and Lichnérowicz

[L],

^{keroc is a} reductive linear

algebraic

group. As X is ^not

unifuled,

^{ker a must be}

trivial,

^{i.e. a is}

injective.

As

Lie(Aut°(X)) = H°(Tx) generically

spans

Tx,

^{it follows} that there is a

subgroup

of the translation _group on

Alb(X) acting

^{on X} with Zariski open

orbit,

^{hence X itself is a torus}

(and

^{so X =}

Alb(X)).

Suppose

^{next that x =}

^1,

and let S be a

generic

^{fibre of} (p. Then

K(S) =

⁰

by

litaka’s fibration theorem

([U]

^Thm.

5.10,

p.

58)

^{and moreover the}

2-forms on X

yield

^{a nonzero 2-form on}

^S,

^{so that S must be an abelian or}

K3 surface. In the former case there is

nothing

^{to prove; in} the latter case

S ^must contain a rational

curve Y,

^and

by (5.1)

and observation

(iii)

^above

Y must move on X in a

2-parameter family,

^{which is}

obviously impossible.

If x ⁼ 2 then ₉ is an

elliptic fibration,

^so

again

^{there is}

nothing

^to prove.

It remains to prove that if x ⁼ 3 then K is

ample.

^{If not}

then,

^{as mK is}

free,

there is a curve Y - X with Y. K ⁼ 0. As

above,

^{Y must move in a} 2-dimensional

family

^{and fill} up ^a surface E such that

g(E)

^{is a}

point,

^and

in

particular

^{K (D}

OE

is torsion. Let

g : ~ E

be the

normalization,

^and

É°

^c E the smooth part. ^{Then we have an} inclusion

But as

OE(E)

^{must be}

negative,

^{we have}

hence

by normality

so that

H0(03C90)

⁼

H0(03A920)

⁼

^0, contradicting

^{the existence}

of many

^2-forms

on X and

completing

^the

proof

^{in case n = 3.}

Suppose

^{now n 4. If 0}

x(X)

^{n, we} may

apply

^{induction to a}

generic

^fibre of the stable

pluricanonical morphism

^{of X.}

Suppose 03BA(X) =

^{0. As}

above,

^we may ^{assume K} is

trivial,

^and consider a de Rham

decomposition

^{of X. As}

03A92X

^is

generically spanned, clearly

^{X cannot have}

an irreducible

symplectic

factor of dimension 4 nor an

SU(n a 3) factor,

^{and we} may argue ^as above to see X has no K3 factor either. Thus X is a torus and we are done. Hence we may assume X is of

general

type, and it will suffice to derive a contradiction from the existence of a curve

f :

^{Y - X with K . Y = 0}

(actually,

^{the argument we}

give

^{for this works for}

n ⁼ 3 as

well,

^{but is less}

elementary

^{than the}

above).

^{Put E =}

f*Tx,

F ⁼ A2E* ⁼

f*03A92X.

^We claim first that F is trivial. This follows from LEMMA 5.4. Let F be a

generically spanned

^{vector bundle}

of degree

^{0 on}

a curve. Then F is trivial.

Proof. By hypothesis,

^every

quotient

^{bundle of F must have}

nonnegative degree,

^{and this}

implies

^{that F cannot have a nonzero section}

vanishing somewhere,

^{so that}

h°(F) = rk(F)

^{and we have an exact} sequence

Comparing degrees,

^{we see that}

(torsion)

^{= 0.} D 1 claim next that our bundle E must be semi-stable: indeed if

Eo

^{c E}

were a

(locally split)

^{subbundle of}

positive degree,

then either

Eo

^or

E/Eo

must have rank

2,

hence either

deg(A2Eo)

&#x3E; 0 or

deg(A2(E/Eo))

^0.

Since these ^are

respectively

^{a subbundle and a}

quotient

^{bundle of the trivial}

bundle

A2E,

^{we have a} contradiction.

Using

the theorem of Narasimhan and Seshadri

[NS],

^our semi-stable bundle E is

S-equivalent

^{to a} bundle which arises from a

unitary

^represen-

tation

By assumption,

^the

composite homomorphism

is

trivial,

^{and as}

ker(039B2) = {± 1}

^{it follows that}

im(03C1) ~ {

^±

11. By composing f

^{with a 2 : 1}

covering ^{of Y,}

^we may ^assume p is trivial. This means that E possesses ^a filtration

LEMMA 5.5. Let E be a vector bundle on a curve such that A2E is trivial and

E/E’

^is

trivial for

^some proper subbundle E’ ^c E. Then E is trivial.

Proof.

We may ^assume rkE’ = rkE - 1. Note that E’ = E’ Q

(EIE’)

^{is a} quo- tient of

A2E,

hence E’ is trivial

by

Lemma 5.4. Now take an

arbitrary

^rank-1

trivial subbundle M g E’. Then from the exact sequence

and Lemma 5.4

again,

^{it follows that}

A2(E/M)

^is

^trivial,

^{hence so is M Q}

(E/

M)

⁼

E/M

^hence

E/M ~ E/E’ splits.

^This

implies

^{that the} extension element

[E]

^E

H1«E/E’)*

(D

E’)

⁼

H1(E’)

goes ^{to zero} in

H1((E/E’)*

Q

(E’/

M))

⁼

H1(E’/M).

^M

being arbitrary

^it follows that

[E]

^{= 0 so E} is trivial. D Now a section of the trivial bundle E ⁼

f*TX yields

^{a first order} deformation

f of f,

^and

by

^{a similar} argument ^{as above we see that}

*TX

is trivial as well

etc.

(use

^{the fact that a} deformation of a trivial bundle E

inducing

^{a trivial}

deformation on A2E and

det(E)

^is

trivial).

^Therefore

by applying

^Theorem

1.1(i)

^to

Def(f/X)

^{we see that}

f

^has unobstructed deformations whose

images

fill _up

X,

^{which is a} contradiction. Il

Acknowledgement.

^{1 am indebted to Prof. Dr. H.} Flenner who

suggested

ⁱⁿ

August

1990 that the results of

[R1]

^and

[R2]

^{should be}

susceptible

^{to an}

abstract

approach,

albeit different from the one we take in Sect.

1;

^{and to} my thesis advisor Prof. A.

Ogus who,

^{back in}

1976,

^called my attention ^to Bloch’s

semi-regularity

theorem with the idea

of extending

^{it to char.} p

using crystalline cohomology. Ogus’

^idea

remains,

^to my

knowledge, unexplored and,

^thanks

e.g. ^to Mori

theory,

^{even more} attractive

though,

^{1 have to}

admit,

^{all the more}

over my ^own head.

Finally,

^{thanks are due to}

thé (last)

referee for a

quick

^and

thorough job.

References

[B] Bloch, ^S. Semi-regularity ^{and de Rham} cohomology, ^{Invent. math. 17} (1972), ^51-66.

[D] Deligne, ^{P. Théorème de Lefschetz et critères de} dégénérescence ^{de suits} spectrales, ^Publ.

Math. IHES, 35 (1968), ^197-226.

[F] Fujiki, ^{A. On} automorphism groups of compact ^Kähler manifolds, Invent. Math. 44 (1978),

225-258.

[L] Lichnérowicz, ^A. Sur les transformations analytiques ^des variétés Kähleriennes. C.R. Acad.

Sci. Paris 244 (1957), ^3011-3014.

[NS] Narasimhan, ^M. S., Seshadri, C. S. Stable and unitary ^{vector bundles on a} compact ^Riemann surface. Ann. of ^{Math. 82} (1965), ^540-567.

[R1] Ran, ^Z. Deformations of manifolds with torsion or negative canonical bundle. J. Algebraic Geometry, ¹ (1992), ^279-292.

[R2] Ran, Z. Lifting of cohomology ^and deformations of certain holomorphic maps. Bull. AMS 26 (1992), ^113-117.

[R3] Ran, ^Z. Hodge theory and the Hilbert scheme. J. Differential Geometry ³⁷ ( 1993), 191-198;

Erratum &#x26; Addendum (to appear).

[R4] Ran, ^{Z. The structure} of Gauss-like _maps. Compositio ^Math. 32 (1984), ^171-177

[R5] Ran, Z. Deformations of _{maps. In:} Algebraic ^{curves and} projective geometry, ^E. Ballico, ^C.

Cilberto, eds., ^{Lecture notes} in Math 1389. Berlin: Springer ^1989.

[R6] Ran, Z. Deformations of Calabi-Yau Kleinfolds. In: Essays ^{on mirror} manifolds, S. T. Yau ed. _pp. 451-457. Hong Kong: International Press 1992.

[R7] Ran, Z. Canonical infinitesimal deformations. Preprint.

[S] Schlessinger, ^M. Functors of Artin rings, ^{Trans. AMS 130} (1968), ^208-222.

[U] Ueno, ^K. Classification theory ^of algebraic varieties and compact complex spaces. Lecture notes in Math. 439. Berlin: Springer ^1975.

[V] Voisin, C. Sur la stabilité des sous-variétés lagrangiennes des variétés symplectiques holomorphes. Preprint.

[Z] Zariski, O. Theorem of Riemann-Roch for high multiples ^{of an} effective divisor on an

algebraic ^{surface. Ann. Math. 76} (1962), ^560-615.